3 Separation of Variables
The free particle Schrodinger Eigen equation in natural units can be written in the form
\begin{equation} (\partial _{xx}+\partial _{yy}-E)\psi = Q_{free}\psi = 0\\ \end{equation}
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Where \(Q_{free}\) is a linear operator on the \(L^2\) space of complex scalar functions \(\psi : \mathbb {R}^2 \rightarrow \mathbb {C}\), denoted \(\mathcal{H}\) -with the usual inner product. Then we consider operators \(L\) which leave the solution space invariant.
\begin{equation} L = a(x,y)\partial _x+b(x,y)\partial _y+c(x,y) \end{equation}
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Such that
\begin{equation} [L,Q] = R_L(x,y) Q \end{equation}
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Plugging in \(L\) and \(Q\)
\begin{equation} [L,Q] = \end{equation}
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